A little time after,the two new friends were running in the snow, happily jumping over those punny confused humans that were looking at them with surprised faces. A dinosaur joined them for a little, but he wasn't used to the cold winter... and so he quickly went back to jumping cactus in the desert.
Time passed, and evening came. The gray pony had to return to their mail-delivery work.
"Can we play again tomorrow?", they asked. "Sure" replied the unicorn "but I wont remember you."
"Why not?" asked the pony "Did I do something to make you angry?".
"An evil ---dev--- wizard put a curse on me. You will have to find a way and break the curse. Unlock my powers and we will be able to play forever".
.... And so a new quest had started for the gray pony.

So, apparently there are actually two ways to play the Hat Dash game:
- the regular intended mode: just wear one of the mask hats on any site. After a little wait you shall receive a notification that you earned the privilege to play the game. From that point on, the game button should be on the Bash site

- the "Easter Egg" way. Just input the Konami code on the site (up up down down left right left right B A [Return]). This unlocks the game but only until you change page. Apparently this was made so that people who don't log on the site can play too. Why someone who doesn't have an account should be on the Bash site and think to use the Konami code there I don't know.
There is a caveat though:Catija confirmed here my initial fear.
@SPArcheon If you haven't unlocked the game through the hat, scores aren't counted - there's a note that says exactly that so that players using the code understand their scores aren't being saved. The code is actually there so that anyone can play, including people who aren't logged in. -
While all the scores, including those got BEFORE unlocking the game through the hat are indeed recorded, the leaderboard simply ignores them.
Therefore, I strongly suggest that if you are actually interested in trying to climb the leaderboard, you should not lose your time with the Easter Egg Konami code unlocked version but instead unlock the permanent version of the game and only play that
As a bonus, if you really wish to see the code doing something, try it AFTER you unlocked the game either way. Inputing the code a second time will get you a nice Troll video. While we are at it, the same will happen with every "secret" url you may remember from last year, so don't try unless you want to increase Yaakov special troll counter (sorry, Yaakov, your secret was never safe with me :P).
......
At least all url but one.
The one that just brings you to a "Stay Tuned..." message. If that one means something only time will tell. For now, Yaakov was quite good at avoiding a clear reply :P
@Derpy stay tuned itself is not a clue to anything other than the simple meaning of the sentence. can't say anything though for any hidden characters that may or may not be returned as well.
@Νеvеrꭑoꭇе good things come to those who wait
UPDATE FOR HISTORICAL RECORD:
At the time of posting this answer the url "/unicorn" was indeed part of a yet-to-be-implemented feature, therefore Yaakov had it just return a placeholder message. Later, that same url was modified and became part of a "quest" to get the Unicorn Defender hat (reference here)